We propose sum rules of x-ray magnetic circularly polarized emission (XMCPE) at $L$ edges for $3d$ transition metals. By making use of combinations of incident and emitted photon helicities, $z$-component expectation values of spin, orbital, magnetic dipole, and quadrupole terms can be obtained separately. The fundamental difference in the sum rules between x-ray magnetic circular dichroism and XMCPE arises from the variety of electron transitions involving core states split by the spin-orbit interaction. The additional electron transition in XMCPE causes complicated angular dependence of the sum rule relation for the spin moment. Our findings promote future $L$-edge XMCPE measurements, which have not been observed at present.

• Received 24 May 2021
• Accepted 23 August 2021

DOI:https://doi.org/10.1103/PhysRevB.104.094419